Exam Problem: calculating the overall distribution based on individual parts' distributions?

Reply Fri 13 May, 2022 03:29 am
Hi guys, hope you are enjoying the day!

So I am having this problem where I have a population of people that participated in a national examination process (N=20,500). The examination required each student to take 4 different tests A, B, C and D and then they would be graded on a scale of [0-20] for each test and an average of the four would form the final grade on a scale of 0 to 20. When the examination process was completed the competent autority uploaded the statistics showing what percentage of the students had scored in each grade category (0-5), (5-10), (10-11), (11-12), ..., (18-19) and (19-20) for every different test. No statistics where uploaded regarding the final score.

And here comes my question: I know in which grade bracket I belong for every of the four tests. Is it possible to calculate approximately in what percentile my overall score would be - based on the distributions of each of the four tests?
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Reply Fri 13 May, 2022 09:48 am
Off the top of my head, I would say no. The problem you have is that the four tests are not mutually independent. A candidate who does very well on one test will probably do very well on all four tests. Because the results are not independent, you can't just add the means and compute a standard deviation and expect it to be accurate.
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Reply Sat 14 May, 2022 10:44 pm
You know the number of students with each score in all four tests.

Therefore, notwithstanding that the results are not independent, identically distributed random variables, it is "possible to calculate approximately" a percentile estimate.

One approximation would be the average percentage of all the scores that are lower than your average.

If you really wanted to crack the nut with a sledge hammer then calculate the 20,000 plus population's probability generating function for each test.
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Reply Wed 27 Sep, 2023 02:58 am
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